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Q. 6.35

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An Introduction to Thermal Physics
Found in: Page 246
An Introduction to Thermal Physics

An Introduction to Thermal Physics

Book edition 1st
Author(s) Daniel V. Schroeder
Pages 356 pages
ISBN 9780201380279

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Short Answer

Verify from Maxwell speed distribution that the most likely speed of a molecule is 2kTm.

The most likely speed of a molecule is vmp=2kTm.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1. Given information

Maxwell speed distribution function is given by

Dv=m2πkT324πv2e-mv22kT..........................(1)

Step 2. Calculation

Differentiate both sides of equation (1) with respect to v.

role="math" localid="1646971259346" dDvdv=ddvm2πkT324πv2e-mv22kT=m2πkT324π2ve-mv22kT-2mv32kTe-mv22kT=8πm2πkT32ve-mv22kT1-mv22kT............................(2)

At most likely speed of the gas molecules, the derivative of the Maxwell speed distribution function will be zero.

Substitute vmp for v and 0 for dDvdv into equation (2) and simplify to obtain the most likely speed.

0=8πm2πkT32ve-mv22kT1-mv22kT1-mvmp22kT=0 [As v≠0]vmp=2kTm

Most popular questions for Physics Textbooks

Show explicitly from the results of this section that G=Nμ for an ideal gas.

In this problem you will investigate the behavior of ordinary hydrogen, H2, at low temperatures. The constant ε is0.0076eV. As noted in the text, only half of the terms in the rotational partition function, equation 6.3, contribute for any given molecule. More precisely, the set of allowed j values is determined by the spin configuration of the two atomic nuclei. There are four independent spin configurations, classified as a single "singlet" state and three "triplet" states. The time required for a molecule to convert between the singlet and triplet configurations is ordinarily quite long, so the properties of the two types of molecules can be studied independently. The singlet molecules are known as parahydrogen while the triplet molecules are known as orthohydrogen.

(a) For parahydrogen, only the rotational states with even values of j are allowed.Use a computer (as in Problem 6.28) to calculate the rotational partition function, average energy, and heat capacity of a parahydrogen molecule. Plot the heat capacity as a function of kT/t

(b) For orthohydrogen, only the rotational states with odd values of j are allowed. Repeat part (a) for orthohydrogen.

(c) At high temperature, where the number of accessible even-j states is essentially the same as the number of accessible odd-j states, a sample of hydrogen gas will ordinarily consist of a mixture of 1/4 parahydrogen and 3/4 orthohydrogen. A mixture with these proportions is called normal hydrogen. Suppose that normal hydrogen is cooled to low temperature without allowing the spin configurations of the molecules to change. Plot the rotational heat capacity of this mixture as a function of temperature. At what temperature does the rotational heat capacity fall to half its hightemperature value (i.e., to k/2 per molecule)?

(d) Suppose now that some hydrogen is cooled in the presence of a catalyst that allows the nuclear spins to frequently change alignment. In this case all terms in the original partition function are allowed, but the odd-j terms should be counted three times each because of the nuclear spin degeneracy. Calculate the rotational partition function, average energy, and heat capacity of this system, and plot the heat capacity as a function of kT/t.

(e) A deuterium molecule, D2, has nine independent nuclear spin configurations, of which six are "symmetric" and three are "antisymmetric." The rule for nomenclature is that the variety with more independent states gets called "ortho-," while the other gets called "para-." For orthodeuterium only even-j rotational states are allowed, while for paradeuterium only oddj states are allowed. Suppose, then, that a sample of D2 gas, consisting of a normal equilibrium mixture of 2/3 ortho and 1/3para, is cooled without allowing the nuclear spin configurations to change. Calculate and plot the rotational heat capacity of this system as a function of temperature.*

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(b) Repeat the calculation for a hydrogen molecule (H2) and for a helium atom, and discuss the implications.

(c) Escape speed from the moon's surface is only about 2.4 km/s. Explain why the moon has no atmosphere.

Consider an ideal gas of highly relativistic particles ( such as photons or fast-moving electrons) whose energy-momentum relation is E=pc instead of E=p22m. Assume that these particles live in a one-dimensional universe. By following the same logic as above, derive a formula for the single particle partition function, Z1, for one particle in the gas.

Consider a large system of N indistinguishable, noninteracting molecules (perhaps an ideal gas or a dilute solution). Find an expression for the Helmholtz free energy of this system, in terms of Z1, the partition function for a single molecule. (Use Stirling's approximation to eliminate the N!.) Then use your result to find the chemical potential, again in terms of Z1.
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